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The prime radical of the special Lie algebras and the elementary Chevalley groups. (English) Zbl 1099.17013
The paper under review studies the prime radicals of Lie algebras, associative rings and groups, and their relationships. Recall that a Lie algebra \(L\) is special if it has an associative enveloping \(R\) that satisfies a polynomial identity. One of the main results of the paper under review, Theorem 1.18, deals with special Lie algebras in characteristic 0. Let \(L\) be such an algebra and \(R\) an associative enveloping PI algebra of \(L\). If \(M\) is an ideal of \(L\) then \(\text{Rad}(M)\) coincides with \(C(M,\text{Rad}(R))\) and with \(M\cap C(L,\text{Rad}(R))\). Here Rad is the prime radical, and \(C(A,B) = \{a\in A\mid [a,b]\in B, b\in B\}\).
Furthermore the author computes the prime radicals of matrix Lie algebras (Theorem 3.1), and establishes a formula for the prime radicals of the subgroups of the general linear group over the rationals (Theorems 2.2 and 2.4).

17B60 Lie (super)algebras associated with other structures (associative, Jordan, etc.)
16N60 Prime and semiprime associative rings
20G05 Representation theory for linear algebraic groups
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